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Properties

Label 2-326700-1.1-c1-0-116
Degree $2$
Conductor $326700$
Sign $-1$
Analytic cond. $2608.71$
Root an. cond. $51.0755$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·7-s + 7·13-s + 7·19-s − 4·31-s − 37-s − 5·43-s + 18·49-s + 61-s + 5·67-s + 10·73-s + 4·79-s − 35·91-s + 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.88·7-s + 1.94·13-s + 1.60·19-s − 0.718·31-s − 0.164·37-s − 0.762·43-s + 18/7·49-s + 0.128·61-s + 0.610·67-s + 1.17·73-s + 0.450·79-s − 3.66·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 326700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 326700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(326700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(2608.71\)
Root analytic conductor: \(51.0755\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 326700,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.95412719965603, −12.41050002287465, −11.99770362613099, −11.43094508274840, −11.05916527815333, −10.48599276647634, −10.10276377992143, −9.541542344599910, −9.306395003570393, −8.777362321760363, −8.355876872695148, −7.684691858577129, −7.235941102439814, −6.626867942176225, −6.403849211611591, −5.841298313359651, −5.476974930267060, −4.879993734468873, −3.990029358971533, −3.540142549799227, −3.416949100832096, −2.805126335922175, −2.084579396299092, −1.247185356843009, −0.8043841076366568, 0, 0.8043841076366568, 1.247185356843009, 2.084579396299092, 2.805126335922175, 3.416949100832096, 3.540142549799227, 3.990029358971533, 4.879993734468873, 5.476974930267060, 5.841298313359651, 6.403849211611591, 6.626867942176225, 7.235941102439814, 7.684691858577129, 8.355876872695148, 8.777362321760363, 9.306395003570393, 9.541542344599910, 10.10276377992143, 10.48599276647634, 11.05916527815333, 11.43094508274840, 11.99770362613099, 12.41050002287465, 12.95412719965603

Graph of the $Z$-function along the critical line